Behaviour of Dirichlet \(L\)-Function of Principal Character at \(1\)

Theorem

For principal Dirichlet character \(\chi_0\) modulo \(q\), we have that

\[ \lim_{s \to 1^+} (s - 1) L(s, \chi_0) = \frac{\varphi(q)}{q}.\]

and hence

\[ \lim_{s \to 1^+} L(s, \chi_0) = \infty.\]

That is \((Ls, \chi_0)\) has a simple pole at \(1\) with residue \(\frac{\varphi(q)}{q}\).

Proof

This follows from this formula and the behaviour of the Riemann zeta function. In particular we have

\[\begin{align*} \lim_{s \to 1^+} (s - 1) L(s, \chi_0) &= \lim_{s \to 1^+} \zeta(s) \prod_{p \mid q} \left(1 - \frac{1}{p^s}\right) \\ &= \lim_{s \to 1^+} (s - 1) \zeta(s) \prod_{p \mid q} \left(1 - \frac{1}{p^s}\right) \\ &= \lim_{s \to 1^+} \left((s - 1) \zeta(s) \right) \lim_{s \to 1^+} \left(\prod_{p \mid q} \left(1 - \frac{1}{p^s}\right) \right) \\ &= \prod_{p \mid q} \left(1 - \frac{1}{p}\right) \\ &= \frac{1}{q} \left(q \prod_{p \mid q} \left(1 - \frac{1}{p}\right)\right) \\ &= \frac{\varphi(q)}{q} \\ \end{align*}\]

noting the Euler totient function formula.

Then clearly because \(s - 1 \to 0\) as \(s \to 1^+\) it would induce a contradiction for \(L(s, \chi_0)\) to be bounded as \(s \to 1^+\).